How can we measure cosmological constant if we can't measure ground state energy? From what I understand, we can only measure energy differences (see for example Peskin & Schroeder page 21, last paragraph), and therefore the ground state of a system cannot really be measured.
From what I also understand, when scientists first measured the cosmological constant, they thought it would be the vacuum energy of the universe (which turned out not to be the case, at least so far). My question is: why did we think that we were able to measure the vacuum energy of the universe? Isn't this just the ground state energy that we can’t measure?
 A: In general relativity you can measure the energy of the vacuum. The vacuum energy corresponds to a term in Einstein's equations proportional to $\Lambda$
$$
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu} = \frac{8\pi G_N}{c^2} T_{\mu\nu}
$$
Essentially, gravity sees all forms of energy, even a constant energy density you would otherwise ignore.
In quantum field theory, there is no way to predict the value of $\Lambda$. The current approach is to fix the value of the cosmological constant based on observations. There's no problem with this.
The issue arises in thinking about how we would derive various contributions to $\Lambda$ if we had access to a high-energy "UV-complete" theory. Then, naive estimates suggest that this UV theory could only reproduce the observed value we see if there is a cancellation between different contributions to $\Lambda$ to a huge number of decimal places. No one has a fully convincing explanation to this problem, but on the other hand there is no mathematical contradiction implied by the problem that must be resolved, and there's nothing that stops cosmologists from doing cosmology by setting $\Lambda$ to its observed value.
